The present paper studies the limiting behavior of the average score of a sequentially selected group of items or individuals, the underlying distribution of which, F, belongs to the Gumbel domain of attraction of extreme value distributions. This class contains the Normal, Lognormal, Gamma, Weibull and many other distributions. The selection rules are the “better than average” (β = 1) and the “β-better than average” rule, defined as follows. After the first item is selected, another item is admitted into the group if and only if its score is greater than β times the average score of those already selected. Denote by Y̅k the average of the k first selected items, and by Tk the time it takes to amass them. Some of the key results obtained are: under mild conditions, for the better than average rule, Y̅k less a suitable chosen function of logk converges almost surely to a finite random variable. When 1 − F(x) = e−[xα + h(x)], α > 0 and h(x) / xα →x → ∞ 0, then Tk is of approximate order k2. When β > 1, the asymptotic results for Y̅k are of a completely different order of magnitude. Interestingly, for a class of distributions, Tk, suitably normalized, asymptotically approaches 1, almost surely for relatively small β ≥ 1, in probability for moderate sized β and in distribution when β is large.